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84
Expander Flows, Geometric Embeddings and Graph Partitioning
 IN 36TH ANNUAL SYMPOSIUM ON THE THEORY OF COMPUTING
, 2004
"... We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a ..."
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Cited by 238 (18 self)
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We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in , whose proof makes essential use of a phenomenon called measure concentration.
The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into ℓ1
 In Proceedings of the 46th IEEE Symposium on Foundations of Computer Science
, 2005
"... In this paper we disprove the following conjecture due to Goemans [16] and Linial [24] (also see [5, 26]): “Every negative type metric embeds into ℓ1 with constant distortion.” We show that for every δ>0, and for large enough n, there is an npoint negative type metric which requires distortion atl ..."
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Cited by 125 (10 self)
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In this paper we disprove the following conjecture due to Goemans [16] and Linial [24] (also see [5, 26]): “Every negative type metric embeds into ℓ1 with constant distortion.” We show that for every δ>0, and for large enough n, there is an npoint negative type metric which requires distortion atleast (log log n) 1/6−δ to embed into ℓ1. Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot [19], establishing a previously unsuspected connection between PCPs and the theory of metric embeddings. We first prove that the UGC implies superconstant hardness results for (nonuniform) SPARSEST CUT and MINIMUM UNCUT problems. It is already known that the UGC also implies an optimal hardness result for MAXIMUM CUT [20]. Though these hardness results depend on the UGC, the integrality gap instances rely “only ” on the PCP reductions for the respective problems. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of UNIQUE GAMES. Then, we “simulate ” the PCP reduction and “translate ” the integrality gap instance of UNIQUE GAMES to integrality gap instances for the respective cut problems! This enables us to prove a (log log n) 1/6−δ integrality gap for (nonuniform) SPARSEST CUT and MINIMUM UNCUT, and an optimal integrality gap for MAXIMUM CUT. All our SDP solutions satisfy the socalled “triangle inequality ” constraints. This also shows, for the first time, that the triangle inequality constraints do not add any power to the GoemansWilliamson’s SDP relaxation of MAXIMUM CUT. The integrality gap for SPARSEST CUT immediately implies a lower bound for embedding negative type metrics into ℓ1. It also disproves the nonuniform version of Arora, Rao and Vazirani’s Conjecture [5], asserting that the integrality gap of the SPARSEST CUT SDP, with the triangle inequality constraints, is bounded from above by a constant.
Euclidean distortion and the Sparsest Cut
 In Proceedings of the 37th Annual ACM Symposium on Theory of Computing
, 2005
"... BiLipschitz embeddings of finite metric spaces, a topic originally studied in geometric analysis and Banach space theory, became an integral part of theoretical computer science following work of Linial, London, and Rabinovich [29]. They presented an algorithmic version of a result of Bourgain [8] ..."
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Cited by 93 (20 self)
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BiLipschitz embeddings of finite metric spaces, a topic originally studied in geometric analysis and Banach space theory, became an integral part of theoretical computer science following work of Linial, London, and Rabinovich [29]. They presented an algorithmic version of a result of Bourgain [8] which shows that every
Solving Euclidean Distance Matrix Completion Problems Via Semidefinite Programming
, 1997
"... Given a partial symmetric matrix A with only certain elements specified, the Euclidean distance matrix completion problem (IgDMCP) is to find the unspecified elements of A that make A a Euclidean distance matrix (IgDM). In this paper, we follow the successful approach in [20] and solve the IgDMCP by ..."
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Cited by 69 (14 self)
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Given a partial symmetric matrix A with only certain elements specified, the Euclidean distance matrix completion problem (IgDMCP) is to find the unspecified elements of A that make A a Euclidean distance matrix (IgDM). In this paper, we follow the successful approach in [20] and solve the IgDMCP by generalizing the completion problem to allow for approximate completions. In particular, we introduce a primaldual interiorpoint algorithm that solves an equivalent (quadratic objective function) semidefinite programming problem (SDP). Numerical results are included which illustrate the efficiency and robustness of our approach. Our randomly generated problems consistently resulted in low dimensional solutions when no completion existed.
On Lagrangian relaxation of quadratic matrix constraints
 SIAM J. Matrix Anal. Appl
, 2000
"... Abstract. Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equ ..."
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Cited by 50 (19 self)
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Abstract. Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equivalent to semidefinite programming relaxations. For several special cases of QQP, e.g., convex programs and trust region subproblems, the Lagrangian relaxation provides the exact optimal value, i.e., there is a zero duality gap. However, this is not true for the general QQP, or even the QQP with two convex constraints, but a nonconvex objective. In this paper we consider a certain QQP where the quadratic constraints correspond to the matrix orthogonality condition XXT = I. For this problem we show that the Lagrangian dual based on relaxing the constraints XXT = I and the seemingly redundant constraints XT X = I has a zero duality gap. This result has natural applications to quadratic assignment and graph partitioning problems, as well as the problem of minimizing the weighted sum of the largest eigenvalues of a matrix. We also show that the technique of relaxing quadratic matrix constraints can be used to obtain a strengthened semidefinite relaxation for the maxcut problem. Key words. Lagrangian relaxations, quadratically constrained quadratic programs, semidefinite programming, quadratic assignment, graph partitioning, maxcut problems
Semidefinite Programming and Integer Programming
"... We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems. ..."
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Cited by 48 (7 self)
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We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems.
Embeddings of negativetype metrics and an improved approximation to generalized sparsest cut
 IN SODA ‘05: PROCEEDINGS OF THE SIXTEENTH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 2005
"... In this paper, we study the metrics of negative type, which are metrics (V,d) such that √ d is an Euclidean metric; these metrics are thus also known as “ℓ2squared” metrics. We show how to embed npoint negativetype metrics into Euclidean space ℓ2 with distortion D = O(log 3/4 n). This embedding re ..."
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Cited by 44 (0 self)
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In this paper, we study the metrics of negative type, which are metrics (V,d) such that √ d is an Euclidean metric; these metrics are thus also known as “ℓ2squared” metrics. We show how to embed npoint negativetype metrics into Euclidean space ℓ2 with distortion D = O(log 3/4 n). This embedding result, in turn, implies an O(log 3/4 k)approximation algorithm for the Sparsest Cut problem with nonuniform demands. Another corollary we obtain is that npoint subsets of ℓ1 embed into ℓ2 with distortion O(log 3/4 n).
On the MatrixCut Rank of Polyhedra
 Mathematics of Operations Research
, 2001
"... Lov'asz and Schrijver (1991) described a semidefinite operator for generating strong valid inequalities for the 01 vectors in a prescribed polyhedron. Among their results, they showed that n iterations of the operator are sufficient to generate the convex hull of 01 vectors contained in a poly ..."
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Cited by 31 (0 self)
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Lov'asz and Schrijver (1991) described a semidefinite operator for generating strong valid inequalities for the 01 vectors in a prescribed polyhedron. Among their results, they showed that n iterations of the operator are sufficient to generate the convex hull of 01 vectors contained in a polyhedron in nspace. We give a simple example, having Chv'atal rank 1, that meets this worst case bound of n. We describe another example requiring n iterations even when combining the semidefinite and GomoryChv'atal operators. This second example is used to show that the standard linear programming relaxation of a kcity traveling salesman problem requires at least bk=8c iterations of the combined operator; this bound is best possible, up to a constant factor, as k + 1 iterations suffice. Key words. Semidefinite programming, integer hull, rank of polytopes, cutting planes, projection operators. Many structures in combinatorial optimization can be modeled as a set of 01 vectors in...
Cuts for mixed 01 conic programming
, 2005
"... In this we paper we study techniques for generating valid convex constraints for mixed 01 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 01 linear programs, such as the Gomory cuts, the liftandproject cuts, and cuts from other hierarchies of ti ..."
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Cited by 29 (0 self)
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In this we paper we study techniques for generating valid convex constraints for mixed 01 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 01 linear programs, such as the Gomory cuts, the liftandproject cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 01 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 01 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.